(adds quotes) NEW YORK, Sept 6 - Shinobu Asagoe, the last unseeded player left in the women's singles, reached her first grand slam quarter-final with a 7-6 4-6 6-3 win over Greece's Eleni Daniilidou at the U.S. Open on Monday. The 28-year-old, who beat two seeds to reach the last 16, raced into a 5-0 lead in the decider after the opening two sets had been closely contested. But she was suddenly gripped with nerves as Daniilidou, also seeking a first appearance in the quarter-finals of a major, produced a desperate fightback. The Greek battled back to 5-3 and threatened to break Asagoe's serve again before she drifted a backhand volley beyond the baseline to spark wild celebrations from the Japanese player. Asagoe will be a huge underdog when she plays the winner of the last 16 match between former winners Venus Williams and Lindsay Davenport but she said she will not be intimidated. "I'm not really satisfied right now, I have to go further from here," she said. "I'm happy but I know I have more matches to play. I will have to be at 120 percent in the next round."